Does nuclear norm decrease as some elements in a matrix are set zero?

Question

Let $M$ be a generic nonzero real matrix and $M_{0}$ is constructed by replacing some elements in $M$ with zero. Let $||\cdot||_{*}$ denotes the nuclear norm operator. Is it true that $||M||_{*}\geq ||M_{0}||_{*}$ regardless of the replacement rule?

Answer 1

No. Consider $$ A=\pmatrix{1&1&1\\ 1&1&1\\ 1&1&1},\ B=\pmatrix{1&0&1\\ 0&1&1\\ 1&1&1}. $$ Since $A$ is positive semidefinite, it nuclear norm is equal to its trace, which is $3$. The matrix $B$ is symmetric but indefinite. Hence its nuclear norm is the sum of absolute values of all its eigenvalues, i.e. $|1|+|1+\sqrt{2}|+|1-\sqrt{2}|=1+2\sqrt{2}>3$.

P.S. I asked a similar question before about the induced $2$-norm (and the answer is also negative).